Kahn’s Principle and the Semantics of Discrete Event Systems

Xiaojun LiuEE290N Class ProjectDecember 10, 2004

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Kahn Process Networks

An elegant model of computation for deterministic concurrent processes.

Processes communicate streams of data asynchronously through unbounded FIFO channels.

A denotational semantics based on complete partial orders (CPOs) and Scott continuous functions.

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Kahn’s Principle

The input-output function of any network is denoted by the least fixed point solution of the network’s recursion equations.

Robert Yates and Guang Gao, “A Kahn Principle for Networks of Nonmonotonic Real-Time Processes”http://www.sable.mcgill.ca/~hendren/ftp/acaps/memo60.ps.gz

p qx

z

y

(y, z) = f(x)

F(f)(x) = (p(x, 2(f(x))), q(1(f(x))))

f = fix(F)

Can we extend this to timed systems?

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Previous Work: Yates & Gao Timed stream: a sequence of time-stamped

tokens. The time stamps are strictly increasing and

separated by a minimum time interval 0. Processes are -causal.

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Previous Work, Continued

The principle novelty of the new functional (F) … is that it cuts off all outputs as soon as the earliest new output token appears.

The new functional lets the network “run” at each step only until the earliest new output token appears.

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Highlights of Approach

A direct extension of KPN to DE systems.

More general than the previous work outlined earlier.

Useful guidance to develop operational semantics and simulator.

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Prefix Order − Definition

Let T, a poset, be the set of all tags. Let L(T) be the set of lower sets of T.

A signal is a function from a lower set LL(T) to some value set V,

signal: L V A signal s1: L1 V is a prefix of s2: L2 V,

denoted s1 s2, if and only if

L1 L2, and s1(t) = s2(t), tL1

Let S(T, V) be the set of all signals from lower sets of T to V.

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Prefix Order − Examples

Given any set A, treated as a poset with the discrete order, S(A, B) is the set of all partial functions from A to B.

Let T = [0, ), and V = V {}, where represents the absence of value, S(T, V) is the set of DE signals.

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Prefix Order − Properties

For any poset T of tags and set V of values, S(T, V) with the prefix order is a poset a CPO a complete lower semilattice (i.e. any

subset of signals have a GLB) For any network of processes that are

Scott continuous functions from input to output, Kahn’s principle applies.

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Discrete Event Signals

The set of DE signals, S([0, ), V {}) is a CPO.

Examples:

clock:

0 1 2 3 …

zeno:

0 1 2 3 …

…

½

r-zeno:

0 1 2 3 …

…

½

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Discrete Event Processes

adds1: L1 V

s2: L2 V

s: L V

L = L1 L2

s(t) = s1(t) + s2(t), tL

delayby 1

s1: L1 V s2: L2 V

L2 = L1{1} [0, 1)s2(t) = s1(t 1), when t 1 and

, when t [0, 1)

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Biased Merge or “Default”

biasedmerge

s1: L1 V

s2: L2 V

s: L V

Let p = sup L2.If pL2, L = L1 L2 {q | q p and t[p, q], s1(t)V }and if pL2, L = L1 L2 {q | q p and t(p, q], s1(t)V }s(t) = s1(t), when s1(t)V

s2(t), otherwise, tL

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A Discrete Event System

biasedmerge

delayby 1

zeno

0 1 2 3 …½

…

0 1½

0 1½

…

0 1 2½

…

0 1 2½

… …

0 1 2 3 …½

… …

0 1 2 3 …½

… … …

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YADES

biasedmerge

lookaheadby 1

r-zeno

0 1 2 3 …

…

½

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Continuity and Causality

A process may be continuous but not causal, e.g. “lookahead by 1”.

A process may be causal but not continuous, e.g. one that produces an output event after counting an infinite number of input events.

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Discrete Signals

A DE signal s: L V is discrete if s-1(V) is order-isomorphic with a lower set

of Nand s-1(V) is a finite set, or L = [0, sup s-1(V)).

Let D([0, ), V {}) be the set of discrete signals. With the prefix order, it is a poset, CPO, and a complete lower semilattice.

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?

A Caveat

biasedmerge

clock:0 1

0 1½

…

zeno:

0 1½

…

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Operational Semantics

Represent signals as timed streams Processes map input timed streams to

output timed streams An alternative development of Chandy

and Misra’s distributed DE simulation

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Representing Discrete Signals as Timed Streams Only “present” events?

(0.0, 1)many-to-one

(0.0, 1), (1.0, 1)

prefix

not a prefix

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Null Events

(0.0, 1), (1.0, )

one-to-many

(0.0, 1), (1.0, 1), (2.0, )

(0.0, 1), (2.0, )

(0.0, 1), (1.0, ), (2.0, )

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Simultaneous Events

Extend the tag set to [0, ) N, with the lexicographical order.

All developments so far can be generalized.

Any continuous/causal DE process at a time t can be treated as a Scott continuous function from input sequences to output sequences.

A “snapshot” of a DE system at a time t is a KPN.

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Current Status

Substantial progress on theoretical development

Developing a plan for implementation Introduce “end-of-stream” marker in KPN Change event queue to contain

sequences of events with the same time stamp and receiver

Key criterion for improvement: more efficient simulation due to less traffic into/out of the event queue